A Lattice-Theoretical Framework for Annular Filters in Morphological Image Processing

We study the idempotence of operators of the form ?∨id∧δ (where ?≤δ and both ? and δ are increasing) on a modular lattice ?, in relation to the idempotence of the operators ?∨id and id∧δ. We consider also the conditions under which ?∨id∧δ is the composition of ?∨id and id∧δ. The case where δ is a dilation and ? an erosion is of special interest. When ? is a complete lattice on which Minkowski operations can be defined, we obtain very precise conditions for the idempotence of these operators. Here id∧δ is called an annular opening, ?∨id is called an annular closing, and ?∨id∧δ is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.